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310                       ANALYTICAL MECHANICS
where Ie is the moment of inertia of the pendulum about an axis through the center of mass parallel to (lie axis of vibration and K is the corresponding radius of gyration.
SECOND APPROXIMATION. — Starting with t lie energy equation we have
, -»gh
= mgD (cos 6 — cos a),
,.      f~l    '           de
or       dt = \/ — — - • V 2 mgD
. mgD   VCos 6> - cos a
r  /, 1 o • o0~
COS (? = 1 — 2 Sill2
L                                2.
„ a        .   ,»c nr -r — sin- .
Integrating the left-hand member of the last equation between the limits t= 0 and 1= . and indicating tin* integration of the right-hand member between the corresponding limits we have
The last integral cannot be evaluated in a finite number of terms, but we can expand the integrand into a power series, every term of which is intcgrable.
Let                       sin • •• = sin ^ sin
2               2i
Then $ = 0 when 0=0, and <£ = ^ when 0 = a ; further
2 sin- cos $